SL(2,R)/SO(2,r)=H^2(2-dim hyperbolic space)

the questions below for scientific debate
1.how to understand SL(2,R)/SO(2,r)=H^2(2-dim hyperbolic space)
2.Let M be a compact manifold ,T compact lie group,what can we say about
M^T(the set of fixed points of M under the action of T),is M^T manifold?
how to argue?further,if T is torus,and M is T-set,how to see